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Theorem un0.1 44799
Description: is the constant true, a tautology (see df-tru 1543). Kleene's "empty conjunction" is logically equivalent to . In a virtual deduction we shall interpret to be the empty wff or the empty collection of virtual hypotheses. in a virtual deduction translated into conventional notation we shall interpret to be Kleene's empty conjunction. If 𝜃 is true given the empty collection of virtual hypotheses and another collection of virtual hypotheses, then it is true given only the other collection of virtual hypotheses. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
un0.1.1 (      ▶   𝜑   )
un0.1.2 (   𝜓   ▶   𝜒   )
un0.1.3 (   (      ,   𝜓   )   ▶   𝜃   )
Assertion
Ref Expression
un0.1 (   𝜓   ▶   𝜃   )

Proof of Theorem un0.1
StepHypRef Expression
1 un0.1.1 . . . 4 (      ▶   𝜑   )
21in1 44591 . . 3 (⊤ → 𝜑)
3 un0.1.2 . . . 4 (   𝜓   ▶   𝜒   )
43in1 44591 . . 3 (𝜓𝜒)
5 un0.1.3 . . . 4 (   (      ,   𝜓   )   ▶   𝜃   )
65dfvd2ani 44603 . . 3 ((⊤ ∧ 𝜓) → 𝜃)
72, 4, 6uun0.1 44798 . 2 (𝜓𝜃)
87dfvd1ir 44593 1 (   𝜓   ▶   𝜃   )
Colors of variables: wff setvar class
Syntax hints:  wtru 1541  (   wvd1 44589  (   wvhc2 44600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-vd1 44590  df-vhc2 44601
This theorem is referenced by:  sspwimpVD  44939
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